On spinors and their transformation

Transcription

1 AMERICAN JOURNAL OF SCIENTIFIC AND INDUSTRIAL RESEARCH, Science Huβ, htt: ISSN: 5-69X On sinors and their transforation Anaitra Palit AuthorTeacher, P5 Motijheel Avenue, Flat C,Kolkata 77,India ABSTRACT The Dirac sinor and its transforation roerties for an iortant art in the foundation of Relativistic Quantu Mechanics and of course of article hysics. The Dirac sinor is neither a scalar nor a four vector. It has an identity of its own ---- it is a sinor. The article clearly brings out the fact that the sinor coonents transfor individually on the assage fro one reference frae to another though the sinor transforation atri sees to roduce the iression that each coonent in the transfored frae is created by the interaction of the four sinor coonents in the original frae. Such issues have been handled in this article. Keywords: Relativistic Quantu Mechanics Dirac Sinor, Klein-Gordon Equation, Dirac equation, Lorentz-Transforation Matri, Sinor Transforation Matri INTRODUCTION The concet of the Dirac sinor, and its transforation roerties for an iortant entity in the construction of Relativistic Quantu Mechanics and of course of article hysics. The Dirac sinor is neither a scalar nor a four vector. It has its own transforation rules defined by its transforation atri and as such has an identity of its own ---- it is a sinor. Each coonent of the transfored Dirac solution is obtained by the interaction of all the coonents of the original solution and this rocedure is quite different fro the transforation of scalars. A careful reconsideration reveals that the coonent solutions of the Dirac equation transfor individually.. The atter will be confired by roving a new result S(ΛΨ(t,, E, = Ψ[Λ(t, Λ(E, ] ( Where, Λ: Lorentz-Transforation Matri S(Λ: Sinor Transforation Matri,5 It is iortant to note that Ψ[Λ(t, Λ(E, ] is the value of Ψ at (t,,, E = [Λ(t,,Λ(E, ] One ay also envisage the coonent solutions of the Dirac equation to be scalars. This is in view of the fact that the individual coonents of the Dirac solution are also solutions of the Klein-Gordon equation [6,7] and that the Klein Gordon solution is necessarily a scalar. The atter ay be resolved if we assue that the Klein-Gordon Equation has nonscalar solutions aart fro the scalar ones. The Dirac Equation and its Lorentz-Covariance The Dirac equation 8,9 is Lorentz-covariant. In the unried frae it reads: (i γ Ψ (t,, E, = ( In the ried frae it reads: (i γ Ψ ( t,, E, = ( The reserved nature of the equations reveals the fact that they have identical solutions in their resective fraes of reference In articular we ay write, Ψ (t,,e, = Ψ (t,, E, ( (t,,e, on the right hand side and the left hand side reresents different oints in the two reference fraes having identical values of the coordinates. By different oints we ean that they are not the corresonding oints of a Lorentz transforation. This result shall be of iense use to us in the future course of develoing the article. It is a direct consequence of the rincile of relativity. The transforation atri re-eained: Let the function Ψ(t,, E, in the unried frae transfor to Ψ ( t,, E, in the ried frae.thus we have, Ψ (t,, E, = S(ΛΨ(t,, E, (5 And (t,, E, = [Λ (t,, Λ(E, ] (6 In the above relations we cannot assue that Ψ (t,,e, =Ψ[Λ(t,,Λ(E, ] (7 Rather we are defining Ψ by equation (5 so as to reserve the for of the Dirac equation on transforation. At this oint let us define a new transforation in the following way: NT[Ψ(t,, E, ] = Ψ[Λ(t, Λ(E, ] (8

2 A. J. Sci. Ind. Res.,, (: 8-87 In the above relation i.e., (7, NT stands for our new transforation and NT(Ψ is the value of Ψ at (t,, E, = [Λ(t,,Λ(E, ]. We ay use the above transforation to ove inside the sae frae fro one oint to another inside the sae reference frae and deterine the si function at the new location (of the sae frae or we ay use it to ove fro one frae of reference to another. But this is not our sinor transforation. The action of the new transforation has been illustrated in the figure below. Now by alying equation ( we ay clai that Ψ(t,, = Ψ (t,,. Indeed by alying the new transforation we have oved within the unried frae fro (t,, to (t,,.according to equation ( the sinor transfored value at (t,, in the ried frae is equal to the value of the si function at (t,, in the unried frae. Ψ(t,, is the solution of equation ( at (t,, and Ψ (t,, is the solution of equation ( at (t,, and equations ( and ( have identical solutions at the different oints (t,, due to the rincile of relativity. Now, Ψ(t,, E, = Ψ[Λ(t,, Λ(E, ] (9 Ψ (t,,e, = S(ΛΨ(t,, E, ( Therefore, S(ΛΨ(t,, E, = Ψ[Λ(t,, Λ(E, ] ( Fig. The first interretation will be alied in the subsequent discussion. Perforing the eercise: Let us erfor the following eercise. We aly the new transforation to ove fro the oint (t,, to (t,, inside the unried reference frae and calculate the si function at the new location. At the sae tie we aly the S atri on the si-function at (t,, and ass to the ried reference frae. The action has been ortrayed in the diagra below. Fig. In fact we have roved (7 Equation ( is in erfect confority with secific eales of the Dirac Sinor. We get the sae result by alying either side of equation on the Dirac sinor. Suorting Calculations : In the above discussion I have roved that S(ΛΨ(t,, E, = Ψ[Λ(t, Λ( E, ] Where S(Λ is the sinor transforation atri And Λ is the Lorentz Transforation atri. I have claied that both the left hand side and the right hand side of the forula ( roduce the sae results. Let e illustrate this For a article oving along the -ais (of the unried frae with oentu the Dirac sinnor is given by : Ψ( = e( i Let the ried frae be oving wrt the unried frae with seed V along the - direction. Moentu= = Λ( 8

3 A. J. Sci. Ind. Res.,, (: 8-87 The Dirac solution is given by: Ψ ( = E + e( i Indeed we have S(ΛΨ = Ψ[Λ(t,, E, ] and the right had side is ore convenient to use. Now this roble takes on a secial diension when we consider articles at rest.the Dirac solution for such articles is given by: Ψ r = ω r ( e(-iε r t, r=,,, Where, ω, ω, ω and ω are the usual colun atrices containing zeros and ones in roer order. If the conventional si function of a oving sinor(oentu= is known to us the right hand side of forula ( easily redicts the si function of the sinor wrt to a frae where it is at rest. But the other way round there is soe difficulty if the functional deendence on oentu is not known to us. If the for of deendence of the si function on oentu is not visible to the forula Ψ = Ψ[Λ(t, Λ(E, ] it cannot erfor the conversion work. The left hand side sees to offer us soe advantage in this atter. Let us check on this oint. If a coonent of the rest sinor be zero it could either be a scalar zero or it could be of the for f( such f(= for =. A correct knowledge of the situation could be gained by considering the Dirac equation. Let us write the four coonent equations for otion along the -ais( = = i i + ψ = i i + ψ = y z i + i + ψ = i + i + ψ = The above set ay be groued into two identical arts: i i + ψ = ( i + i + ψ = And, i i + ψ = ( i + i + ψ = The second set ie, equations( ay be given trial solutions of the for ψ = ψ = With the first[equations (] set we try, ( e( ψ = f i ( e( ψ = f i The eonential art has been redicted fro the rest sinor solution. Substituting these trial solutions into equations ( we have, f( [ E ] = f( And, [ ] f( = f( Or, [ E ] f( = f( Using the relation, E = We have, 8

4 A. J. Sci. Ind. Res.,, (: 8-87 f( f( = [ ] Writing f( = [ ] We have the noral Dirac Solution for a article oving along the -direction. The above choice of f is such that it satisfies the condition that ( ψ ψ is a Lorentz-invariant Our solution is: T [ ] (,,, e( i ψ = [T stands for transose in the above eression] The solutions for the two-diensional or three diensional otion ay be calculated fro the transforation rules or directly fro the Dirac Equations. TWO DIMENSIONAL MOTION [,, = ] y z The Dirac Equation for two-diensional otion are: + = y i + i + ψ = y i i ψ + + = y i + i + + ψ = y i i ψ Trial Solution: ( (5 ψ = ψ =, ψ= f (, E, e( i, ψ = f(, E, e( i y y Substituting the trial solution into equations ( and using the relation E = + we have, f = f Solution: + i y y + iy ψ = f(,,, e( i If the trial solution is taken as: ψ = ψ =, ψ = f ( E,, e( i, ψ = f( E,, e( i y y The Dirac-solution works out to: iy T ψ = f(,,, e( i The condition that ψ ψ is a Lorentz invariant gives the value of f = [ ] in both the above cases. In the eonential arts we could have used e( i instead of e( i to obtain other tyes of solution. These results confor to the standard ones. Fo transforation we sily relace by the corresonding Lorentz values of and, y y. Our new transforation rule, ψ ( te,,, = ψ [ Λ( t,, Λ ( E, ] indeed reains valid. The three diensional solution ay also be evaluated by using suitable trial solutions. An eleent of arbitrariness: Now the sinor transforation atri is defined by S - γ ν S = Λ ν γ (6 Let us look into the roof of the above relation: We have, (- i γ + Ψ( = ( = Λ ν ν (8 And S - Ψ ( = Ψ( (9 Equation ( ay be written as 8

5 A. J. Sci. Ind. Res.,, (: 8-87 (-i γ Λ ν ν Ψ ( + S - (Λ Ψ ( = ( Left ultilying the above equation by S we have -is Λ ν γ S - ν Ψ ( + Ψ ( = [It is iortant to note that Λ ν are nubers(atri eleents while the gaas are atrices theselves. ] In order to reserve the Dirac equation we ust have, S - γ ν S = Λ ν γ ( But it is esecially interesting to take note of the atri oerator: S (Λ = S e(-i M ω ( Where ω is the boost angle (tanh ω = v c or the rotation angle. Here M is a constant scalar. [Please take note of the fact that e (-i M ω = for ω=. Also take note of the fact that e(-i M ω is a scalar and not a atri.] We shall rove that if the transforation atri S reserves the Dirac equation the atri S (Λ also reserves it.. Now, (S - = [S e (-imω] - = e(imω S ( We have, Ψ( = (S - Ψ ( Under the action of S (Λ equation ( transfors to (-i γ Λ ν ν Ψ ( + e(imω S - (Λ Ψ ( = Then left ultilying by S = S e (imω we obtain, (S (-i γ Λ ν ν Ψ ( + (S - (Λ Ψ ( = Or, -i S e(-imω Λ ν γ e(imωs - ν Ψ ( + Ψ ( = ( Or, -i S Λ ν γ S - ν Ψ ( + Ψ ( = By using (7 that is, Λ ν γ = S - γ ν S, We obtain, (- i γ + Ψ ( = (5 So the Dirac equation is again reserved. Invariance of nor of the si function wrt to S ilies Ψ S S Ψ = Ψ e (imωs S (- imωψ = Ψ e imω(-imωψ = Ψ IΨ = Ψ Ψ Thus the nor is reserved for S also. Indeed S and S reserve the Dirac equation for the sae boost or rotation. In fact the scalar M ay contain eressions containing coonents of the energy-oentu four-vector. Of these several atrices the left hand side of ( chooses a articular one for transforing the sinor at rest and so the transforation aears to be of a unique nature. But the transforation is not actually unique if we consider all the atrices given by (8 Indeed the right hand and the left hand sides of ( have equal robles in handling sinors at rest if the functional deendence of si on the oentu coonents is not known to us. The Klein-Gordon Connection: It is a well known fact that the individual coonents of the Dirac solution satisfy the Klein-Gordon equation. Now we know that the solutions of the K-G equation ust necessarily be a scalar and therefore the individual coonents of the Dirac equation ust also be scalars[if they transfor individually] and hence the four-coonent Dirac solution should be a scalar. But this is not true! The Dirac solution Ψ is not a scalar (si-bar si is a scalar. Therein lays the contradiction! Interestingly the Dirac coonents ay be transfored in two ways: By using the rule Ψ (t,, E, = Ψ[Λ(t,, Λ(E, ].This we have roved in the revious section. In this case the coonents transfor individually. By using the relation Ψ (t,, E, = S(ΛΨ(t,, E,. This is the usually followed rocedure where all the coonents are involved. But if the relation S(Λ[Ψ(t,,E ]=Ψ[Λ(t,, Λ(E, ] is true both the ethods should roduce the sae result. This has been clearly elained in the revious section. By alying the first ethod we conclude that the coonents transfor individually. At the sae tie they should be scalars (each coonent being a solution of the K-G equation. Let us first write the Klein Gordon Equation (natural units have been assued: That is Now, Ψ Ψ = Ψ S S Ψ S S= I ( + ψ = 85

6 A. J. Sci. Ind. Res.,, (: 8-87 The Klein Gordon oerator itself is a Lorentz-invariant quantity. Therefore to reserve the above relationshi (on Lorentz Transforation we have two otions: The function ψ is itself a Lorentz invariant quantity Keeing in ind that the Klein-Gordon oerator is a differential oerator and that the result of the oeration deends not only on the oint transfored but also on the neighboring oints, we coe to an iediate conclusion that other solutions are ossible (The Dirac solution is such a solution.in fact the Klein-Gordon equation is a artial differential equation and it ay have several solutions deending on the boundary conditions. This strengthens the arguent. The Klein Gordon Solution ay not be a scalar! It is iortant that we investigate the solutions other than the two usual tyes- the Dirac sinor and the usual zero-sin solution. CALCULATIONS The Klein Gordon equation (Assuing natural units: c=, h =: ( + ψ = ; Or, ψ + = y z We aly the searation of variables technique. Let ψ = φ (tφ ( φ (y φ (z Substituting the above trial solution in ( we obtain, ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ + = y z The solutions are: ϕ = Ae( ikt + A e( ikt ϕ = Be( ik + B e( ik ϕ = Ce( iky + C e( iky ϕ = De( ikz + De( ikz The searation constants are -----(7 k, k, k and k ψ =Σ const ψψψψ -----(8 General Soln: Suation is carried over all the values of the constants The eleentary Klein Gordon eigen-function ay be obtained for ( by asigning suitable values to the constants. For eale utting A =,B =,C = and D = we obtain one of the eleentary Klein- Gordon solutions. Also we assue, k =E (=ω.the condition k k k k = gets autoatically iosed. We ay identify k, k and k as the oenta, and (since h = The Dirac Coonents: First we take note of the fact that the values of energy and individual coonents of oentu four-vector are constants in the eleentary solutions (eigenfunctions of both the Klein Gordon and the Dirac equations.(in fact to verify the validity of the solutions wrt the equations we artial differentiate the holding each coonent of the energy oentu four-vector constant with resect to t,, y and z searately.therefore these values ay be used as the constants A, B etc. This basic inforation is crucial to what we are to do net. First let us consider the usual Dirac solution for articles oving along the ais with oentu and energy E(Dislayed reviously: For r=, the first Dirac coonent is: ψ ( = [ ] e( i We ay get this result fro equation Set ( by assuing. A = [ ] and k = All other constants are assued to be zero It ay also be noted that E is constant for a articular eigenfunction. But the above function is not a scalar since E is not an invariant. Moreover one aears in the coefficient while the other in the eonent. For r= the fourth coonent is: ψ ( = e( i ( Again we ay obtain the above function fro Set( A by assuing = ( K = and other constants are zero. It ay again be noted that and E being constant fore a articular eigenfunctions ay be fitted into the constants. But the above function is not a scalar since,e and transfor without reserving the function. Siilar stateents ay be ade in reference to other Dirac Coonents. It is difficult to say whether, 86

7 A. J. Sci. Ind. Res.,, (: 8-87 such solutions eist in nature (they are neither zerosin nor half sin articles! but the ossibility should be elored. We could redict other solutions of the Klein Gordon equation by assigning other suitable tyes of values to the constants and elore the ossibility of eistence of such articles! CONCLUSIONS We have roved the iortant relation ( S(ΛΨ(t,, E, = Ψ[Λ(t,, E, ] Where, Λ: Lorentz-Transforation atri S(Λ: Sinor Transforation atri It is iortant to note that in the above Ψ[Λ(t,, E, ] is the value of Ψ at (t,, E, = Λ(t, Λ(E, So the Dirac coonents ay transfor individually or collectively roducing the sae result. It has also been shown that the Klein-Gordon solution is not necessarily a scalar ACKNOWLEDGEMENTS: I eress y sincere indebtedness and gratitude to all authors whose tets have insired e to aintain an active interest in the disciline of hysics. REFERENCES.Walter Greiner, Relativistic Quantu Mechanics, Third Edition, Sringer International Edition, age-..michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantu Field Theory, Levant Books, Kolkata, India, age -9. Franz Schwabl, Advanced Quantu Mechanics, Second Edition, Sringer International Edition age.. Walter Greiner, Relativistic Quantu Mechanics, Third Edition, Sringer International Edition age Franz Schwabl, Advanced Quantu Mechanics, Second Edition, Sringer International Edition Reference [], age - 7. Franz Schwabl, Advanced Quantu Mechanics, Second Edition, Sringer International Edition, age Walter Greiner, Relativistic Quantu Mechanics, Third Edition, Sringer International Edition age Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantu Field Theory, Levant Books, Kolkata, India, age -9, age -.. Walter Greiner, Relativistic Quantu Mechanics, Third Edition, Sringer International Edition, age 6 87